Existence and concentrate behavior of positive solutions for Chern–Simons–Schrödinger systems with critical growth

Abstract: In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schrodinger equation \begin{equation} -\Delta u+u +\lambda\left(\int_{0}^{\infty}\frac{h(s)}{s}u^{2}(s)ds+\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}\right)u=f(x,u)+\epsilon k(x)\quad\quad \text{in}\,\,\mathbb{R}^2, \\ \end{equation} where $h(s)=\int_{0}^{s}\frac{l}{2}u^{2}(l)dl$, $\lambda>0$ and the nonlinearity $f:\mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}$ behaves like $\text{exp}(\alpha\vert u\vert^{2})$ as $\vert u\vert\rightarrow \infty$. For the case $\epsilon=0$, we can get a mountain-pass type solution.

Abstract: In this paper, we study the following Chern-Simons-Schrodinger system − Δ u + u + ( h 2 ( | x | ) | x | 2 + ∫ | x | ∞ h ( s ) s u 2 ( s ) d s ) u = f ( u ) in R 2 , where h ( s ) = 1 2 ∫ 0 s r u 2 ( r ) d r and the nonlinearity f ∈ C ( R , R ) . If f satisfies asymptotically 3-linear at infinity, we establish the existence of ground state solutions by using general minimax principle. Moreover, we obtain the multiplicity of solutions by a mountain pass approach introduced by Hirata, Ikoma and Tanaka (2010) [7] .

Abstract: In this paper, we consider the equation $$\begin{aligned} -\varepsilon ^{2}\Delta u+ V(x)u+\left( A_{0}(u)+A_{1}^{2}(u)+A_{2}^{2}(u)\right) u=f(u) \ \ \ \ \mathrm {in} ~ H^{1}({\mathbb {R}}^{2}), \end{aligned}$$ where $$\varepsilon $$ is a small parameter, V is the external potential, $$A_i(i=0,1,2)$$ is the gauge field and $$f\in C({\mathbb {R}}, {\mathbb {R}})$$ is 5-superlinear growth. By using variational methods and analytic technique, we prove that this system possesses a ground state solution $$u_\varepsilon $$ . Moreover, our results show that, as $$\varepsilon \rightarrow 0$$ , the global maximum point $$x_\varepsilon $$ of $$u_\varepsilon $$ must concentrate at the global minimum point $$x_0$$ of V.

Abstract: In this paper, we investigate the following Chern-Simons-Schrodinger system { − Δ u + V ( x ) u + A 0 u + A 1 2 u + A 2 2 u = f ( x , u ) , x ∈ R 2 , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 u 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 u 2 , ∂ 2 A 0 = − A 1 u 2 , where V is the potential, ∂ 1 = ∂ ∂ x 1 , ∂ 2 = ∂ ∂ x 2 for x = ( x 1 , x 2 ) ∈ R 2 , A j : R 2 → R is the gauge field ( j = 0 , 1 , 2 ) and the nonlinearity f ( x , s ) ∈ C ( R 2 × R , R ) behaves like e 4 π s 2 as | s | → + ∞ . If V and f are both asymptotically periodic at infinity, we prove the existence of positive ground state solutions by combining the Nehari manifold methods with the Trudinger-Moser inequality.

Abstract: In this paper, we study the following Chern–Simons–Schrodinger equation $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$ where $$\omega ,\lambda >0$$ and $$h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$$ . Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $$\lambda \rightarrow 0$$ .

Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for

Abstract: 1. The Main Result; Examples . . . . . . . . . . . . . . . . . . . . . . . 316 2. Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 319 3. The Constrained Minimization Method . . . . . . . . . . . . . . . . . . 323 4. Further Properties of the Solution . . . . . . . . . . . . . . . . . . . . 328 5. The \"Zero Mass\" Case . . . . . . . . . . . . . . . . . . . . . . . . . 332 6. The Case of Dimension N = 1 (Necessary and Sufficient Conditions) . . . . . 335 Appendix. Technical Results . . . . . . . . . . . . . . . . . . . . . . . . 338

Abstract: Abelian Nonrelativistic Model.- Nonabelian Nonrelativistic Model.- Abelian Relativistic Model.- Nonabelian Relativistic Model.- Quantum Aspects.

Abstract: We construct a nonrelativistic field theory for the second-quantized N-body system of point particles with Chern-Simons interactions. Various properties of this model are discussed: its obvious and hidden symmetries, its relation to a relativistic field theory, and its supersymmetric formulation. We present classical, static solutions--solitons--that satisfy a self-dual equation, which is equivalent to the Liouville equation; hence, it is completely solvable. The dynamical role of the Chern-Simons interaction is demonstrated: the interaction does not merely change statistics but also provides the forces that bind the classical solitons.

Abstract: (1992). Nontrivial Solution of Semilinear Elliptic Equations with Critical Exponent in R. Communications in Partial Differential Equations: Vol. 17, No. 3-4, pp. 407-435.